Nobody had succeeded to answer the question “Is there a perfect magic
cube of order 5? No one knows”asked and popularized by Martin Gardner in
Scientific American (1976) and in his book Time Travel and Other Mathematical
Bewilderments (1988).

In November 2003, with
my German friend Walter Trump, we constructed the
first perfect magic cube of order 5, the
smallest possible perfect cube.

First perfect magic cube of order 5, by
Trump & Boyer, 2003.This
cube contains all the integers from 1 to 53 = 125.
There are 109 ways to get the magic sum 315, with its 25 rows,
25 columns, 25 pillars, 4 triagonals, and 30 diagonals.It is
impossible to construct a smaller perfect magic cube.(click
on the image to enlarge it)

More than three centuries
before, in 1640, Pierre de Fermat sent Mersenne a nearly perfect magic cube
of order 4, with 64 magic lines: Fermat made a mistake announcing 72 magic lines in his
cube, but 64 remains an excellent result compared to the needed 76 theoretical magic
lines of a perfect magic cube of order 4 that we know now to be impossible.

Some explanations about the construction method used for our cube. We
first computed a large number of auxiliary cubes of order 3. These
auxiliary cubes are "central symmetrical", meaning that all their
13 alignments of 3 numbers going through the center have x + 63 + y = 189.
These auxiliary cubes have some other partial magic characteristics: 29
of the 49
different possible alignments (including 14 of the 18 diagonals and all
4 triagonals) have already the
same sum = 189. You can check these characteristics, looking at the central
cube of order 3 included in our magic cube of order 5. Then, using
these auxiliary cubes, we tried to fill, always by computer, the 53
- 33 = 98 missing numbers mainly using complementary numbers
x + 189 + y = 315. That's why you can see a lot of different symmetries in
our cube.

We ran five computers for several weeks before finding the first
solution. With this method, it was very easy to find cubes of order 5 with
26 magic diagonals. More than 1500 cubes with 28 magic diagonals were
found before finding our first cube with 30 magic diagonals. We used more
than 80,000 different auxiliary cubes of order 3 before finding this first
solution.

Download the perfect
cube of order 5 constructed November 13th 2003 by Walter Trump and Christian
Boyer (Excel
file of 43Kb).

See the numerous
articles published in the world after our discovery. Many thanks
to all these magazines and these web sites. Many thanks also to Martin
Gardner who at age 89 sent us in December 2003 some kind words
from his home in Oklahoma, now scanned with emotion after his death in 2010:

Download
the PDF file (955Kb) of the article published in French in La Recherche.

Two months before this cube of order 5, Walter Trump constructed the first perfect cube of order
6, using in
its center an auxiliary cube of order 4.

Download the perfect
cube of order 6 constructed September 1st 2003 by Walter Trump (Excel
file of 45Kb).

The terminology used in this web site for the "perfect" multimagic cubes,
is the most commonly used: a perfect magic
cube is a magic cube with the additional property that each square is
magic (every diagonal is magic, not just the triagonals).

Definition used for example by Martin Gardner (Scientific American,
January 1976), by William H. Benson and Oswald Jacoby (Magic Cubes: New Recreations,
1981), and also by Clifford A. Pickover (The Zen of Magic Squares, Circles
and Stars, 2002). It is also used in Eric Weisstein’s World of Mathematics
at http://mathworld.wolfram.com/PerfectMagicCube.html.

For multimagic cubes, it means for example that each square of the 32nd-order perfect bimagic
cube is bimagic, and that each square of the 256th-order perfect trimagic cube
is trimagic.

There is another definition of a "perfect" cube, with supplemental properties of
pandiagonality to get, defined by John R. Hendricks. The
above "perfect" cubes are then no more called perfect, but "diagonal"
cubes. See the Perfect
Magic Cube page nicely done by Harvey D. Heinz.

(*) These cubes have supplemental characteristics of pandiagonality
(broken diagonals and/or broken triagonals also magic).(**) This cube has supplemental multimagic
characteristics
(again perfect magic when its numbers are squared or cubed or raised to
the 4th power).(1) And also
India: Frost was English, but was
at that time a missionary in Nasik, India.(2) And also Germany:
Frankenstein was born in Germany, and was 2-years old when his family moved
to Cincinnati, USA.(3) Li Wen sent us this cube in Dec. 2003. He had never published it before.
Li says that he had built it in 1988.

First perfect cube of order 7

The first perfect cube published is of order
7, and was constructed by Rev. A. H. Frost, M.A. of St. John's College,
Cambridge. Frost had been a missionary in a city named Nasik in India.
Thus, his cube is called a "Nasik cube", and was published in 1866,
in an English
scientific magazine, The Quarterly Journal of Pure and Applied Mathematics.

In his Time Travel book, Martin Gardner thought
that the first perfect cube of order 7 had been published by Harry Langman in
1962, in his book Play Mathematics (Hafner Publishing Company). Gardner
was wrong, since we have found Frost's cube constructed in 1866, nearly
one century before. Harry
Langman was a Doctor of Philosophy of Columbia University, and has been Professor
of Mathematics and Department Chairman in three educational institutions.

In this original text, you will read the easy construction method used. Harry
Langman was very modest: "We shall content ourselves with giving
an illustration of a seven-cube". We will replace by: "We shall delight
ourselves..."

First perfect cube of order 8

The first published perfect cube of order 8 was constructed by Gustavus
Frankenstein. It was published in an old American local newspaper, The Cincinnati
Commercial, March 11th, 1875. So this old newspaper is very difficult
to find and locate. And, as far as we know, the associated text has never been
republished in any other book or article. Good news, we have found the original
newspaper,

In this original text, you will read what G. Frankenstein said about
his cube: "This discovery gives me greater satisfaction than
if I had found a gold mine under my door-sill; and it is delight like this that
makes poverty sweeter than the wealth of Craesus."

And he continued: "Perhaps the thing has been done before." We
are nearly sure, dear Gustavus, that the thing had not been done
before for the order 8! But sorry, the thing has been done before for the order
7.

This Cincinnati Commercial reference is reported for example in:

Theory of Magic Squares and of Magic Cubes, by F.A.P. Barnard, page
244 (Memoirs of the National Academy of Sciences, 1888)

Thirteen years after Frankenstein, another American, Frederick
A.P. Barnard, publishes a pandiagonal perfect magic cube of order 8, with
two other cubes of order 11. See
these cubes in the page dedicated to pandiagonal
perfect magic cubes.

First perfect cube of order 9

The first published perfect cube of order 9 was constructed by Charles
Planck, M.A., M.R.C.S., England. It was published in his booklet, The Theory
of Path Nasiks, 1905, greatly inspired by Frost's work.

Download Planck's perfect cube of order 9 (zipped
Excel file of 100Kb, created by Harvey Heinz)

The 4k+2 orders are often very difficult to construct, both for magic squares
and cubes.

The first perfect cube of order 10 has
been received in December 2003. Li Wen, China, says
that he had constructed it in 1988, but he never published it. We already knew
Li Wen for his excellent pentamagic square of order 729.

The first published perfect cubes of order 11 were published in 1888 by Frederick
A.P. Barnard, President of Columbia College (now Columbia
Univ.), New-York , USA. His two cubes of order 11 have even more characteristics: they
are pandiagonal perfect magic cubes.